A345173 Numbers whose multiset of prime factors is separable but has no alternating permutation.

270, 378, 594, 702, 918, 1026, 1242, 1566, 1620, 1674, 1750, 1998, 2214, 2268, 2322, 2538, 2625, 2750, 2862, 3186, 3250, 3294, 3564, 3618, 3834, 3942, 4050, 4125, 4212, 4250, 4266, 4482, 4750, 4806, 4875, 5238, 5454, 5508, 5562, 5670, 5750, 5778, 5886, 6102

Offset: 1

Author

Gus Wiseman, Jun 13 2021

Keywords

• nonn

Comments

A multiset is separable if it has an anti-run permutation (no adjacent parts equal).

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The terms together with their prime indices begin:
   270: {1,2,2,2,3}
   378: {1,2,2,2,4}
   594: {1,2,2,2,5}
   702: {1,2,2,2,6}
   918: {1,2,2,2,7}
  1026: {1,2,2,2,8}
  1242: {1,2,2,2,9}
  1566: {1,2,2,2,10}
  1620: {1,1,2,2,2,2,3}
  1674: {1,2,2,2,11}
  1750: {1,3,3,3,4}
  1998: {1,2,2,2,12}
  2214: {1,2,2,2,13}
  2268: {1,1,2,2,2,2,4}
  2322: {1,2,2,2,14}
		

Crossrefs

The partitions with these Heinz numbers are counted by A345166.

Permutations of this type are ranked by A345169.

Numbers with a factorization of this type are counted by A348609.

A000041 counts integer partitions.

A001250 counts alternating permutations, complement A348615.

A003242 counts anti-run compositions.

A025047 counts alternating compositions, ascend A025048, descend A025049.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344606 counts alternating permutations of prime indices with twins.

A344740 counts twins and partitions with an alternating permutation.

A345164 counts alternating permutations of prime factors.

A345165 counts partitions without an alternating permutation.

A345170 counts partitions with an alternating permutation.

A345192 counts non-alternating compositions, without twins A348377.

A348379 counts factorizations with an alternating permutation.

Cf. A001222, A071321, A316524, A335126, A344614, A344616, A344652, A344653, A345163, A345168, A345193, A347706, A348380, A348613.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
sepQ[y_]:=!MatchQ[y,{_,x_,x_,_}];
Select[Range[1000],Select[Permutations[primeMS[#]],wigQ]=={}&&!Select[Permutations[primeMS[#]],sepQ]=={}&]

Formula

Equals A345171 /\ A335433.